Every FFL will probably be different.
My local SOT is all set-up. He did my live scan prints and photo which he keeps on file. And will sell you a copy for $40 if you want. We fill out the Form 4 together on his computer, I pay the $200, get the "submitted" email, and wait. Once I get the "Approved" email, I go in, fill out my 4473, and take home my can. If I order a can he doesn't have, I go in when it arrives (after waiting on the Form 3 transfer to him), do the Form 4 paperwork like above, and pay his transfer fee.
My FFL sends in my LEO notification for me (not sure if paper or electronic?). My buddy was told to mail in his own after he did his Form 4 at his FFL in another State.
I've gotten 5 cans in the last month and half. I'm waiting longer for Form 3 transfers to my FFL, than Form 4 approvals. I'm averaging about a week on my approvals.